What do we mean by 'Equivalent Projective representation"?by S_klogW Tags: None 

#1
Dec1812, 11:04 AM

P: 19

I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(1)=R'.
But what do we mean by equivalent projective rerpesentations? I've heard of the theorem that the SO(3) group has only 2 inequivalent projective representations. But what does that exactly mean? I am very interested in projective representation because it's projective representation rather than ordinary representation that represents symmetry in Quantum Mechanics since the vector A and exp(id)A represent the same physical state. So does anyone know if there are some books that can serve as an introduction to projective representations? 



#2
Dec2112, 06:27 PM

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P: 8,995

Since you haven't received any replies, I will mention that "Geometry of quantum theory" by Varadarajan covers projective representations and their relevance to quantum mechanics. I hesitate to recommend it because I find it very hard to read, but I don't know a better option.




#3
Dec2112, 07:51 PM

Mentor
P: 16,591

I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first
[tex]Z=\{cI_n~\vert~c\in \mathbb{R}\}[/tex] Then we define a projective representation as a group homomorphism [tex]\rho: G\rightarrow GL_n(\mathbb{R})/Z[/tex] This last group is often called [itex]PGL_n(\mathbb{R})[/itex], or the projective general linear group. Given, [itex]\rho,\rho^\prime[/itex] projective representations, it would make sense to define them equivalent if there exist [itex]U\in O_n(\mathbb{R})/Z[/itex] such that [tex]\rho(g)=U\cdot \rho^\prime(g)\cdot U^{1}[/tex] for all [itex]g\in G[/itex]. The complex case is similar. 


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